Competition between algae and fungi in a lake: a mathematical model

in Science and Engineering, CMMSE 2016 4–8 July, 2016.

Competition between algae and fungi in a lake: a

mathematical model

Iulia Martina Bulai1 and Ezio Venturino1

1 _{Dipartimento di Matematica “Giuseppe Peano”, Universit`a di Torino,}

via Carlo Alberto 10, 10123 Torino, Italy

emails: iuliam@live.it, ezio.venturino@unito.it

Abstract

In this paper a mathematical model for handling water pollution is introduced. We assume that algae and fungi are in competition for resources that come from wastewater. Both algae and fungi need dissolved oxygen (DO) for their biologial process of growth. But there is a difference, indeed algae produce it too and in a higher quantity than the one they use. It is shown that if the coexistence equilibrium exists, it is stable without additional conditions. If the competition rate between algae and fungi is not high for a chosen set of parameters the stability of the coexistence equilibrium is reached even without an external constant input of DO in the system.

Key words: mathematical model, algae, fungi, competition, wastewater MSC 2000: AMS codes (optional)

1

Introduction

Algae are important in a lake, they can improve the quality of the aquatic ecosystem. Under right conditions such as adequate nutrients (mostly phosphorus, but nitrogen is important too) they grow. The nutrients that are present in the wastewater can derive from agricultural and/or industrial discharges. Fungi can be used for biodegradation of organic pollutant in a waterbody, and they grow using the nutrients obtained from the biodegradation, [1]. Some mathematical models in the literature study the behavior of algae biomass in a waterbody in the presence of organic pollutants, [4, 5]. In [2, 3] the case of fungi has been addressed. In this paper we want to study what happens when both algae and fungi are present in the same waterbody, for example in a lake. Furthermore we suppose that they are in competition for the resources coming from the pollutants.

2

The mathematical model

In this paper we introduce a mathematical system that models the behavior of algae and fungi in a waterbody. The waterbody considered could be nutrient-rich waters, like mu-nicipal wastewater or some industrial effluents. Both algae and fungi can feed on these wastes and therefore purify the water, while also producing a biomass suitable for biofuels production. Thus algae and fungi are in competition for food, since both share the same resources. Further, fungi as well as algae need DO to thrive but we assume that the algae’s production and input of DO into the system is much larger than their own use for their growth.

The model consists of three equations that describe the time evolution of the algae population, the fungi population and the DO respectively. The model, in which all the parameters are nonnegative, reads:

dA dt = rAA− aAA− bAA 2_{− cAF (1)} dF dt = hOF k + kOO − aF F − bFF2− cAF dO dt = qO+ gA− aOO− f hOF k + kOO .

In the first equation algae grow at a constant rate rA and are washed out at a constant rate aA. We assume that algae are in competition among themselves at a constant rate bA and

also experience interspecific competition with fungi at rate c.

In the second equation the fungi’s growth depends on the presence of DO. They are washed out at rate aF. The intraspecific competition occurs at rate bF while c denotes the

rate of the interspecific competition with the algae population.

The third equation shows the evolution in time of DO. We assume that it is supplied from external sources at rate qO, but a part of it comes from the algae own production at

rate g. We take further into account its washing out, at rate aOand its depletion due to its

assumption by fungi at rate f  1.

3

The qualitative analysis of the model

First of all to find the equilibrium points of the model, we need to solve the system obtained by setting the right hand side of (1) to zero,

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ A(rA− aA− bAA− cF ) = 0 F  hO k + kOO − aF − bF F − cA  = 0 qO+ gA− aOO− f hOF k + kOO = 0. (2)

Further, for the stability analysis, we need to calculate the Jacobian matrix of the system (1) J = ⎡ ⎢ ⎢ ⎢ ⎣ rA− aA− 2bAA− cF −cA 0 −cF −aF − 2bFF − cA + hO k + kOO hkF (k + kOO)2 g − f hO k + kOO −aO− f hkF (k + kOO)2 ⎤ ⎥ ⎥ ⎥ ⎦. (3) Solving (2) we obtain the analytic expression of three equilibrium points. In addition, we prove that two other equilibria exist. We also show that all these points are conditionally locally asymptotically stable, while the coexistence equilibrium is stable if it is feasible.

Proposition 1. The trivial equilibrium point, E₀= (0, 0, 0), exists if

qO= 0. (4)

Furthermore, it is stable if the following condition holds:

rA < aA. (5)

Proof. For A = F = O = 0 in the system (2) we get that E₀ exists if qO = 0. The

characteristic polynomial associated to the matrix (3) evaluated at E₀ is det(J− μ) = (rA− aA− μ)(−aF − μ)(−aO− μ) = 0.

To have the stability of E₀all the eigenvalues shoud be negative thus the condition (5) must hold.

Proposition 2. The fungi-and-algae-free point E₁ =0, 0, qOa−1_O



exist always. It is stable if the following conditions hold:

rA < aA and

hqO kaO+ kOqO

< aF. (6)

Proof. In fact for A = F = 0 in (2) from the last equation we get O = qOa−1_O . While the

characteristic polynomial associated to E₁ is det(J− μ) = (rA− aA− μ)(−aO− μ)  hqO kaO+ kOqO − aF − μ  = 0. To have all the eigenvalues negative the conditions (6) must hold.

Figure 1: The equilibrium E₁ is stably achieved with the parameter values rA = 10.1273, aA = 16.93072, bA = 11.7382, c = 19.3012, h = 1.61592, k = 0.454245, kO = 5.87845, aF = 2.55798, bF = 0.0344478, qO= 2, g = 3.63317, aO= 5.41771, f = 1.

In Figure 1 one can see that for a chosen set of parameters the equilibrium E₁ is stably achieved.

Proposition 3. The fungi-free equilibrium E₂ =  rA− aA bA , 0,qObA+ g(rA− aA) bAaO  is feasible if rA > aA (7) and it is stable if hO₂ k + kOO2 < aF + cA2 (8) hold.

Proof. If F = 0 in the system (2) we get

⎧ ⎪ ⎨ ⎪ ⎩ rA− aA− bAA = 0 F = 0 qO+ gA− aOO = 0. (9)

From the first equation of (9) it follows

A = rA− aA bA

.

Thus for the nonnegativity of the algae population, (7) must hold. From the third equation instead we get the equilibrium value of the oxygen.

O = qObA+ g(rA− aA) bAaO

The characteristic polynomial associated to E₁ is once again easily obtained, det(J− μ) = (−rA+ aA− μ)(−aO− μ)  hO₂ k + kOO2 − aF − cA2− μ  = 0, as well as its eigenvalues

μ₁ = −ra+ aA < 0 μ₂ = −aO< 0 μ₃ = hO2

k + kOO2 − aF − cA2

Requiring μ₃ < 0 we get (8).

In Figure 2 we show that for a chosen set of parameters the stability of the fungi-free equilibrium, E₂, is attained.

Figure 2: The equilibrium E₂is stable for the parameter values rA= 8.90096, aA= 4.14886, bA = 1.62848, c = 0.0691916, h = 11.8402, k = 1.92602, kO = 16.7554, aF = 18.7713, bF = 15.4976, qO= 69.5303, g = 13.847, aO= 19.037, f = 1.

Proposition 4. The algae-free point is in fact a set of multiple equilibria, namely (0, F₃, O₃), (0, F₄, O₄) and (0, F₅, O₅). Of them, only one is feasible, if

O > aFk h− aFkO

, (10)

and it is stable if

Proof. Part 1: existence

For A = 0 the system (2) becomes ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ A = 0 hO k + kOO − aF − bF F = 0 qO− aOO− f hOF k + kOO = 0. (12)

Solving the second equation with respect to F we get

F = O(h− aFkO)− aFk bF(k + kOO)

. (13)

Condition (10) arises by requiring the positivity of the expression (13). Note that the opposite case obtained when h− aFkO< 0, cannot arise,

O < aFk h− aFkO

,

because from it, O < 0 follows, which is impossible.

Substituting the expression (13) for F into the third equation of the system (12) we obtain the following third degree equation in O

aO3+ bO2+ cO + d = 0, (14) with a = −aObFkO2 < 0 b = qObFkO2 − 2aObFkkO− fh2+ f haFkO c = 2qObFkkO− aObFk2+ f haFk d = qObFk2> 0.

Since a < 0 and d > 0 by Descartes’ rule of signs the third degree polynomial (14) in O has at least one positive root. We are able to show that there is exactly one such root. In the next Table the only four possible cases are summarized.

Cases a b c d number of real positive roots

1) − + − + 3 (impossible)

2) − + + + 1

3) − − − + 1

The first case is impossible, in fact assuming that b > 0 and c < 0 we find

aObFk + f h2

kO

< qObFkO− aObFk + f haF <−qObFkO.

But this is a contradiction, because the term in the middle should be less than a negative term (on the right) and greater than a positive one (on the left side).

Thus there is only one positive equilibrium

E₃ =  0,O3(h− aFkO)− aFk bF(k + kOO3) , O3  if O₃ > aFk h− aFkO ,

with O₃ the real positive root of (14).

Part 2: stability

To study the stability of the equilibrium point we evaluate the Jacobian matrix (3) at E₃. The resulting characteristic polynomial is

det(J− μ) = (rA− aA− cF3− μ) # μ2+  aF + 2bFF3+ aO+ f hkF₃ (k + kOO3)2 + − hO3 k + kOO3  μ +  aO+ f hkF₃ (k + kOO3)2  (aF + 2bFF3)− aOhO3 k + kOO3 < = 0. The eigenvalue μ₁ = rA− aA− cF3is negative if (11) holds, while the roots of the quadratic

polynomial in μ are negative with no further conditions. It turns out that both coefficients of the terms of the two lowest degrees in μ are positive. In fact, substituting F₃, (13), for the coefficient of μ we get

 aF + 2hO₃ k + kOO3 − 2(O₃aFkO+ aFk) k + kOO3 + aO+ f hkF₃ (k + kOO3)2 − hO₃ k + kOO3  =  aF + hO₃ k + kOO3 − 2aF + aO+ f hkF₃ (k + kOO3)2  = O3(h− aFkO)− aFk bF(k + kOO3) + aO+ f hkF₃ (k + kOO3)2 = F₃+ aO+ f hkF₃ (k + kOO3)2 > 0.

Similarly, for the constant term, by dividing by aO, denoting by H is a positive term, we

have:

aF + 2bFF3−_{k + k}hO3

OO3 + H > 0.

Figure 3 shows that for a chosen set of parameters the algae-free equilibrium, E₃, is stably achieved.

Figure 3: The equilibrium E₃ is stabile for the parameters rA = 4.85571, aA = 13.2729, bA = 11.6077, c = 12.7024, h = 11.4803, k = 2.3815, kO = 1.11246, aF = 1.40551, bF = 1.95987, qO= 65.3973, g = 16.7907, aO= 15.374, f = 1.

For the coexistence equilibrium point we have the following result

Proposition 5There exists at least one feasible coexistence equilibrium E₄= (A∗, F∗, O∗) if the following three conditions hold:

rA− aA− bAA∗ > 0, bFbA− c2> 0, (aFc + bF)(k + kOO∗)(rA− aA) > chO∗ (15)

and whenever it exists, it is stable.

Proof. To find the conditions for the existence of the coexistence equilibrium point from

the first equation of the system (2) we get

F = rA− aA− bAA c

and substitute it into the remaining two equations. We solve these two equations with respect to A and we match the resulting expressions

A = (aFc + bF)(rA− aA)(k + kOO)− chO

(k + kOO)(bFbA− c2)

= f hO(rA− aA) + c(aOO− qO)(k + kOO)

cg(k + kOO) + f hbAO

.

Thus, we now have the following cubic polynomial in O:

with a₁ = −aOkO2(bFbA− c2) < 0 for bFbA− c2> 0 b₁ = kO(f hc + bFkOg)(rA− aA) + kO(kOqO− 2aOk)(bFbA− c2) + +(aFkO− h)(fhbA+ ckOg) c₁ = k(f hc + 2bFkOg)(rA− aA) + k(2kOqO− aOk)(bFbA− c2) + +chg(2aFkO− h) + aFkhbA d₁ = aFck2g + k2qO(bFbA− c2) + bFk2g(ra− aA) > 0.

Since for bFbA− c2 > 0, a1 < 0 and d1 > 0 the polynomial (16) has at least one positive root O∗ by the Descartes’ rule of signs. For the feasibility of the equilibrium we need to have F∗ > 0 and A∗> 0, providing thus the first and the third conditions in (8).

To study the stability, the characteristic polynomial of (3) evaluated at E₄ gives the cubic equation det(J− μ) = μ3+ Rμ2+ Sμ + P = 0 with R =  bAA∗+ aO+ bFF∗+ f hkF∗ (k + kOO∗)2  > 0 S =  A∗F∗(bAbF − c2) + bFaOF∗+ f hkF∗(bAA∗+ bFF∗) (k + kOO∗)2 + h 2_kF∗_O∗ (k + kOO∗)3  > 0 P =  (bFbA− c2)  aOA∗F∗+ f hkF∗2A∗ (k + kOO∗)2  +bAA ∗_h2_kF∗_O∗ (k + kOO∗)3 + cghkA ∗_F∗ (k + kOO∗)2  > 0.

For bFbA− c2 > 0, which holds by feasibility, the three eigenvalues are negative. Thus E4

is stable whenever it is feasible.

In Figure 4 we show that the coexistence equilibrium is stable for a selected set of parameter values.

In the next Table we summarize the feasibility conditions for the five equilibrium points of model (1).

In Figure 5 for a chosen set of parameters and the same initial conditions changing the value of qO, the constant input of DO in the system, we obtain the stability of the

coexistence equilibrium, E₄ on the left and of E₂ on the right. Thus, starting from the coexistence equilibrium, by decreasing the rate qO at which oxygen is supplied into the

Figure 4: Two possible configurations for the equilibrium E₄, qO = 0 and qO = 1. It is

stable in the following cases. Left: qO = 0, (1.20, 0.30, 0.55) is achieved for the parameters rA = 1, aA = 0.001, bA = 0.8, c = 0.1224, h = 1.1, k = 1, kO = 1, aF = 0.001, bF = 0.8, qO = 0, g = .1, aO = 0.001, f = 1. Right: qO = 1, (1.08, 1.08, 10.15) is obtained for the

parameters rA = 1, aA = 0.001, bA = 0.8, c = 0.1224, h = 1.1, k = 1, kO = 1, aF = 0.001, bF = 0.8, qO= 1, g = .1, aO= 0.001, f = 1.

Eq. Feasibility conditions Stability conditions

E₀ q₀ = 0 rA < aA E₁ none rA < aA and hqO kaO+ kOqO < aF E₂ rA > aA hO₂ kaO+ kOO2 < aF + cA2 E₃ O₃> aFk h− aFkO rA < aA+ cF3 E₄ rA− aA− bAA∗ > 0, bFbA− c2> 0 (aFc + bF)(k + kOO∗)(rA− aA) > chO∗ none

4

Conclusions and future work

A three dimensional, nonlinear mathematical model has been introduced and analysed. In addition to the trivial equilibrium, four additional equilibrium points have been found. Their stability was been completely analysed. For a chosen set of parameters with the same initial conditions we get the stability of the coexistence equilibrium, E₄, both in the absence,

qO = 0, and with full, qO = 1, external oxygen supply, Figure 4. Thus the constant input

Figure 5: Left: the equilibrium E₄ is stable for qO = 30; rA = 19.6445, aA = 1.73234, bA = 11.5828, c = 3.34324, h = 16.676, k = 12.4305, kO = 0.517493, aF = 3.04963, bF = 1.3597, qO = 30, g = 11.5323, f = 0.835718, aO = 5.42261 Right: the equilibrium E₂ at the stability qO = 20 for rA = 19.6445, aA = 1.73234, bA = 11.5828, c = 3.34324, h = 16.676, k = 12.4305, kO= 0.517493, aF = 3.04963, bF = 1.3597, qO= 20, g = 11.5323, f = 0.835718, aO= 5.42261.

system. In fact algae contribution of DO to the system is enough for the fungi utilization. The simulations of Figure 5 instead show that the DO concentration should not drop below a critical threshold, because in such situation the fungi may disappear. Such a loss would be detrimental for the ecosystem.

One of the hypothesis of the model is the competition for food between algae and fungi, but in an indirect way the results indicate that algae help the fungi growth by producing DO.

In our future research we will compare the model introduced here (1) with another one in which the nutrient equation is also considered, as follows:

dA dt = hAN A kA+ kNN − eA A− mAA2 (17) dF dt = k(N, O)F − eFF − mFF 2 dN dt = qN− r hAN A kA+ kNN − sk(N, O)F − eN N dO dt = qO+ gAA− eOO− cOk(N, O)F with k(N, O) = k1N O k₂+ k₃N + k₄O + k₃k₄N O.

Acknowledgments

This work has been partially supported by the projects “Metodi numerici in teoria delle popolazioni” and “Metodi numerici nelle scienze applicate” of the Dipartimento di Matem-atica “Giuseppe Peano” of the Universit`a di Torino.

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